# Upper bound and lower bound

1. Aug 29, 2010

### l46kok

1. The problem statement, all variables and given/known data

1.
f(n) = n - 100
g(n) = n - 200

2.
f(n) = log(2n)
g(n) = log(3n)

n >= 0 in all cases
Find out if f(n) is an upperbound, lowerbound or both of g(n)
2. Relevant equations

3. The attempt at a solution

in case of 1, f(n) has to be an upperbound of g(n) because when graphed together, f(n) has to be an upperbound of g(n).

For 2, solution does not exist at n = 0. Otherwise, f(n) is a lower bound of g(n). Does this mean that f(n) is a lower bound or both?

2. Aug 30, 2010

### HallsofIvy

Staff Emeritus
Too vague. What you should say is "200> 100 so -100> -200 and n- 100> n- 200 for all n. Since f(n)> g(n) for all n, f is an upper bound of g."

Is suspect that should not be "$n\ge 0$ for both cases" but only n> 0 for the second. As you point out, ln(0) is not defined so the problem makes no sense for n= 0.

ln(2)< ln(3) so ln(n)+ ln(2)< ln(n)+ ln(3) for all n> 0. ln(2n)< ln(3n) for all n> 0.